On the relationships between the factors of the upper and lower central series in some non-periodic groups

This paper deals with the mutual relationships between the factor group $G/\zeta(G)$ (respectively $G/\zeta_k(G)$) and $G'$ (respectively $\gamma_{k+۱}(G)$ and $G^{\mathfrak{N}}$). It is proved that if $G/\zeta(G)$ (respectively $G/\zeta_k(G)$) has finite $۰$-rank, then $G'$ (respectively $\gamma_{k+۱}(G)$ and $G^{\mathfrak{N}}$) also have finite $۰$-rank. Furthermore, bounds for the $۰$-ranks of $G', \gamma_{k+۱}(G)$ and $G^{\mathfrak{N}}$ are obtained.

Keywords:

finite rank , torsion-free rank , section $p$-rank , generalized radical group